Matrix Riemann–Hilbert problem

Remarks

This is the vector form of the Riemann–Hilbert problem; for $n = 1$ it is the scalar problem, which is solved in closed form with logarithms and Cauchy integrals. For $n \ge 2$ the scalar method fails, because the values $G(t)$ need not commute.

The general theory rests on a factorization $G = G_+\, \Lambda\, G_-$ on $\Gamma$, where $G_+$ extends holomorphically and invertibly to $D^+$, $G_-$ does the same on $D^-$ including $\infty$, and $\Lambda(t) = \operatorname{diag}(t^{\kappa_1}, \ldots, t^{\kappa_n})$ when $0 \in D^+$. Such a factorization exists for every Hölder continuous invertible $G$, and the integers $\kappa_1 \ge \cdots \ge \kappa_n$, the partial indices, are unique. Their sum is the winding number of $\det G$ about $0$. The number of solutions of the homogeneous problem and the solvability conditions of the inhomogeneous problem are expressed by the partial indices, not by their sum alone. The same structure appears for matrix symbols on the real line in the matrix Wiener–Hopf factorization. No general method computes the partial indices or the factors of a given $G$ in closed form.

Example: if $G$ is diagonal, the problem splits into $n$ scalar problems, and the partial indices are the winding numbers of the diagonal entries. If $G$ is triangular, the problem is solved by a chain of scalar problems.

The problem is linked to Hilbert's twenty-first problem, which asks for a linear system of differential equations with regular singular points and prescribed monodromy. Plemelj gave a positive answer for regular singular systems in 1908 through this problem. Bolibruch showed in 1989 that the version asking for Fuchsian systems has a negative answer in general. Matrix Riemann–Hilbert problems on contours with prescribed jumps are also the basis of the nonlinear steepest descent method for integrable systems and random matrices.